arXiv subjects

Salvatore Tringali

Publications and source records attributed to Salvatore Tringali.

At least 19 recordsLinked to original sources

arXiv2026

On the automorphisms of numerical power monoids

Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb N$ (the non-negative integers under addition). Denote by $\mathcal P_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set. Tringali and Yan [JCTA, 2025] have recently established that $\mathcal P_{\text{fin},0}(\mathbb N)$ has a unique non-trivial automorphism, and conjectured that the automorphism group of $\mathcal P_{\text{fin},0}(H)$ is trivial whenever $H \ne \mathbb N$. We prove this conjecture and, as a byproduct, give a new proof of the Tringali--Yan theorem.

math.NT↗

arXiv2026

Power Semigroups and Two Rigidity Theorems for Groups

Let $\mathcal P(H)$ be the semigroup obtained by endowing the family of all non-empty subsets of a semigroup $H$ with the setwise operation naturally induced by $H$ on its power set, and denote by $\mathcal P_\text{fin}(H)$ the subsemigroup of $\mathcal P(H)$ consisting of all non-empty finite subsets of $H$. We obtain (as a corollary of a theorem of independent interest) that if $H$ is a group and $K$ is a semigroup, then $\mathcal P(H) \cong \mathcal P(K)$ implies $H \cong K$. The finitary analogue of this statement is considerably more difficult, and we prove it only for $H$ an additive subgroup of the rationals. Most notably, the proof of the second result relies, in a rather circuitous way, on a special case of the Evertse--Schlickewei--Schmidt theorem.

math.GR↗

arXiv2026

Power monoids and their arithmetic: a survey

The non-empty finite subsets of a multiplicatively written monoid form a monoid under setwise multiplication. The same holds for finite subsets containing the identity element. Partly due to their unusual arithmetic properties, these structures, generically known as power monoids, have attracted increasing attention in recent years, stimulating new perspectives in the study of factorizations in non-cancellative or non-commutative settings. We survey these developments and briefly review some related aspects.

math.RA↗

arXiv2026

On the automorphisms of the power semigroups of a numerical semigroup

If $H$ is a numerical semigroup (that is, a cofinite subset of the non-negative integers closed under addition), then the non-empty subsets of $H$ form a semigroup $\mathcal P(H)$ under the sumset operation induced by addition in $H$. Moreover, if $0 \in H$, then $\mathcal P(H)$ is a monoid with identity element $\{0\}$, and the family $\mathcal P_0(H)$ of all subsets of $H$ containing $0$ is a submonoid of $\mathcal P(H)$. We show that the automorphism group of $\mathcal P(H)$ is trivial, and the same holds for $\mathcal P_0(H)$ when $0 \in H$. The proofs blend ideas from combinatorics and semigroup theory.

math.NT↗

arXiv2026

Torsion groups and the Bienvenu--Geroldinger conjecture

Equipped with the operation of setwise multiplication induced by a (multiplicatively written) monoid $H$ on its parts, the collection of all finite subsets of $H$ containing the identity element is itself a monoid, denoted by $\mathcal P_{\textrm{fin}, 1}(H)$ and called the reduced finitary power monoid of $H$. One is naturally led to ask whether, for all $H$ and $K$ in a given class of monoids, $\mathcal P_{\textrm{fin},1}(H)$ and $\mathcal P_{\textrm{fin},1}(K)$ are isomorphic if and only if $H$ and $K$ are. The problem originates from a conjecture of Bienvenu and Geroldinger that was recently settled by the authors. Here, we provide a positive answer to the problem in the case where $H$ and $K$ are cancellative monoids, one of which is torsion. In particular, the answer is in the affirmative when $H$ and $K$ are torsion groups. Whether the conclusion extends to arbitrary groups remains open.

math.GR↗

arXiv2025

On global isomorphisms and a closure property of semigroups

Let $S$ be a semigroup (written multiplicatively). Endowed with the operation of setwise multiplication induced by $S$ on its parts, the non-empty subsets of $S$ form themselves a semigroup, denoted by $\mathcal P(S)$. Accordingly, we say that a semigroup $H$ is globally isomorphic to a semigroup $K$ if $\mathcal P(H)$ is isomorphic to $\mathcal P(K)$; and that a class $\mathscr C$ of semigroups is globally closed if a semigroup in $\mathscr C$ can only be globally isomorphic to an isomorphic copy of a semigroup in the same class. We show that the classes of groups, torsion-free monoids, and numerical monoids are each globally closed. The first result extends a 1967 theorem of Shafer, while the last relies non-trivially on the second and on a classical theorem of Kneser from additive number theory.

math.RA↗

arXiv2025

On the arithmetic of power monoids

Given a monoid $H$ (written multiplicatively), the family $\mathcal{P}_{\mathrm{fin},1}(H)$ of all non-empty finite subsets of $H$ containing the identity element $1_H$ is itself a monoid, called the reduced finitary power monoid of $H$, under the operation of setwise multiplication induced by $H$. We investigate the arithmetic of $\mathcal P_{\mathrm{fin},1}(H)$ from the perspective of minimal factorizations into irreducibles, paying particular attention to the potential presence of non-trivial idempotents. Among other results, we provide necessary and sufficient conditions on $H$ for $\mathcal P_{\mathrm{fin},1}(H)$ to admit unique minimal factorizations. Our results generalize and shed new light on recent developments on the topic.

math.RA↗

arXiv2025

The Automorphism Group of the Finitary Power Monoid of the Integers under Addition

Endowed with the binary operation of set addition carried over from the integers, the family $\mathcal P_{\mathrm{fin}}(\mathbb Z) $ of all non-empty finite subsets of $\mathbb Z$ forms a monoid whose neutral element is the singleton $\{0\}$. Building upon recent work by Tringali and Yan, we determine the automorphisms of $\mathcal P_{\mathrm{fin}}(\mathbb Z)$. In particular, we find that the automorphism group of $\mathcal P_{\mathrm{fin}}(\mathbb Z)$ is isomorphic to the direct product of a cyclic group of order two by the infinite dihedral group.

math.CO↗

arXiv2025

The Category of Atomic Monoids: Universal Constructions and Arithmetic Properties

We introduce and investigate the category $\mathsf{AtoMon}$ of atomic monoids and atom-preserving monoid homomorphisms, which is a (non-full) subcategory of the usual category of monoids. In particular, we compute all limits and colimits, showing that $\mathsf{AtoMon}$ is a complete and cocomplete category. We also address certain arithmetic properties of products and coproducts, providing explicit formulas for some fundamental invariants associated with factorization lengths in atomic monoids.

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arXiv2024

Semigroups of ideals and isomorphism problems

Let $H$ be a monoid (written multiplicatively). We call $H$ Archimedean if, for all $a, b \in H$ such that $b$ is a non-unit, there is an integer $k \ge 1$ with $b^k \in HaH$; strongly Archimedean if, for each $a \in H$, there is an integer $k \ge 1$ such that $HaH$ contains any product of any $k$ non-units of $H$; and duo if $aH = Ha$ for all $a \in H$. We prove that the ideals of two strongly Archimedean, cancellative, duo monoids make up isomorphic semigroups under the induced operation of setwise multiplication if and only if the monoids themselves are isomorphic up to units; and the same holds upon restriction to finitely generated ideals in Archimedean, cancellative, duo monoids. Then we use the previous results to tackle a new case of a problem of Tamura and Shafer from the late 1960s.

math.RA↗

arXiv2024

On the isomorphism problem for power semigroups

Let $\mathcal P(S)$ be the semigroup obtained by equipping the family of all non-empty subsets of a (multiplicatively written) semigroup $S$ with the operation of setwise multiplication induced by $S$ itself. We call a subsemigroup $P$ of $\mathcal P(S)$ downward complete if any element of $S$ lies in at least one set $X \in P$ and any non-empty subset of a set in $P$ is still in $P$. We obtain, for a commutative semigroup $S$, a characterization of the cancellative elements of a downward complete subsemigroup of $\mathcal P(S)$ in terms of the cancellative elements of $S$. Consequently, we show that, if $H$ and $K$ are cancellative semigroups and either of them is commutative, then every isomorphism from a downward complete subsemigroup of $\mathcal P(H)$ to a downward complete subsemigroup of $\mathcal P(K)$ restricts to an isomorphism from $H$ to $K$. This solves a special case of a problem of Tamura and Shafer from the late 1960s and generalizes a recent result by Bienvenu and Geroldinger, where it is assumed, among other conditions, that $H$ and $K$ are numerical monoids.

math.RA↗

arXiv2024

On power monoids and their automorphisms

Endowed with the binary operation of set addition, the family $\mathcal P_{{\rm fin},0}(\mathbb N)$ of all finite subsets of $\mathbb N$ containing $0$ forms a monoid, with the singleton $\{0\}$ as its neutral element. We show that the only non-trivial automorphism of $\mathcal P_{{\rm fin},0}(\mathbb N)$ is the involution $X \mapsto \max X - X$. The proof leverages ideas from additive number theory and proceeds through an unconventional induction on what we call the boxing dimension of a finite set of integers, that is, the smallest number of (discrete) intervals whose union is the set itself.

math.CO↗

arXiv2023

A conjecture by Bienvenu and Geroldinger on power monoids

Let $S$ be a numerical monoid, i.e., a submonoid of the additive monoid $(\mathbb N, +)$ of non-negative integers such that $\mathbb N \setminus S$ is finite. Endowed with the operation of set addition, the family of all finite subsets of $S$ containing $0$ is itself a monoid, which we denote by $\mathcal P_{{\rm fin}, 0}(S)$. We show that, if $S_1$ and $S_2$ are numerical monoids and $\mathcal P_{{\rm fin}, 0}(S_1)$ is isomorphic to $\mathcal P_{{\rm fin}, 0}(S_2)$, then $S_1 = S_2$. (In fact, we establish a more general result, in which $S_1$ and $S_2$ are allowed to be subsets of the non-negative rational numbers that contain zero and are closed under addition.) This proves a conjecture of Bienvenu and Geroldinger.

math.CO↗

arXiv2023

On the finiteness of certain factorization invariants

Let $H$ be a monoid, $\mathscr F(X)$ be the free monoid on a set $X$, and $π_H$ be the unique extension of the identity map on $H$ to a monoid homomorphism $\mathscr F(H) \to H$. Given $A \subseteq H$, an $A$-word $\mathfrak z$ (i.e., an element of $\mathscr F(A)$) is minimal if $π_H(\mathfrak z) \ne π_H(\mathfrak z')$ for every permutation $\mathfrak z'$ of a proper subword of $\mathfrak z$. The minimal $A$-elasticity of $H$ is then the supremum of all rational numbers $m/n$ with $m, n \in \mathbb N^+$ such that there exist minimal $A$-words $\mathfrak a$ and $\mathfrak b$ of length $m$ and $n$, resp., with $π_H(\mathfrak a) = π_H(\mathfrak b)$. Among other things, we show that if $H$ is commutative and $A$ is finite, then the minimal $A$-elasticity of $H$ is finite. This yields a non-trivial generalization of the finiteness part of a classical theorem of Anderson et al. from the case where $H$ is cancellative, commutative, and finitely generated (f.g.) modulo units and $A$ is the set $\mathscr A(H)$ of its atoms. We also check that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, f.g. monoid with trivial group of units whose minimal $\mathscr A(H)$-elasticity is infinite.

math.RA↗

arXiv2023

Subcommutativity of integrals and quasi-arithmetic means

Let $(X, \mathscr{L}, λ)$ and $(Y, \mathscr{M}, μ)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with either $0 < λ(A) < 1 < λ(X)$ and $0 < μ(B) < μ(Y)$, or the other way around. In addition, let $I \subseteq \mathbb{R}$ be a non-empty open interval, and suppose that $f,g\colon I \to \mathbb{R}_{+}$ are homeo\-morphisms with $g$ increasing. We prove that the functional inequality $$ f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;dμ\right)\right)d λ\right)\! \le g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;dλ\right)\right)d μ\right) $$ is satisfied by every $\mathscr{L} \otimes \mathscr{M}$-measurable simple function $h: X \times Y \to I$ if and only if $f=a g^b$ for some $a,b \in \mathbb{R}_{+}$ with $b\ge 1$. An analogous characterization is given for probability spaces.

math.FA↗

arXiv2023

A characterization of atomicity

In [Math. Proc. Cambridge Philos. Soc. 64 (1968), 251-264], P.M. Cohn famously claimed that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, a counterexample was provided by A. Grams in [Math. Proc. Cambridge Philos. Soc. 75 (1974), 321-329]: Every commutative domain with the ACCP is atomic, but not vice versa. This has led to the question of finding a sensible (ideal-theoretic) characterization of atomicity. The question (explicitly stated on p. 3 of A. Geroldinger and F. Halter-Koch's 2006 monograph on factorization) is still open. We settle it using the language of monoids and preorders.

math.RA↗

arXiv2023

Factorization under Local Finiteness Conditions

It has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain factorizations, herein called $\preceq$-factorizations, for the $\preceq$-non-units of a (multiplicatively written) monoid $H$ endowed with a preorder $\preceq$, where an element $u \in H$ is a $\preceq$-unit if $u \preceq 1_H \preceq u$ and a $\preceq$-non-unit otherwise. The ``building blocks'' of these factorizations are the $\preceq$-irreducibles of $H$ (i.e., the $\preceq$-non-units $a \in H$ that cannot be written as a product of two $\preceq$-non-units each of which is strictly $\preceq$-smaller than $a$); and it is interesting to look for sufficient conditions for the $\preceq$-factorizations of a $\preceq$-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). This is precisely the kind of questions addressed in the present work, whose main novelty is the study of the interaction between minimal $\preceq$-factorizations (i.e., a refinement of $\preceq$-factorizations used to counter the ``blow-up phenomena'' that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the ``local behaviour'' of the pair $(H, \preceq)$. Besides a number of examples and remarks, the paper includes many arithmetic results, a part of which are new already in the basic case where $\preceq$ is the divisibility preorder on $H$ (and hence in the setup of the classical theory).

math.RA↗

arXiv2022

On the density of sumsets, II

Arithmetic quasi-densities are a large family of real-valued set functions partially defined on the power set of $\mathbb{N}$, including the asymptotic density, the Banach density, the analytic density, etc. Let $B \subseteq \mathbb{N}$ be a non-empty set covering $o(n!)$ residue classes modulo $n!$ as $n\to \infty$ (e.g., the primes or the perfect powers). We show that, for each $α\in [0,1]$, there is a set $A\subseteq \mathbb{N}$ such that, for every arithmetic quasi-density $μ$, both $A$ and the sumset $A+B$ are in the domain of $μ$ and, in addition, $μ(A + B) = α$. The proof relies on the properties of a little known density first considered by Buck in 1946.

math.NT↗